Subgroups — #15781025

Order 3328.

286 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718085	C₂	1	2	yes	maximal
2	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic
3	#13115008	C₂	1	1	yes	maximal, characteristic
4	#13112929	C₂	1	2	yes	maximal
5	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
8	C₄×C₁₃×M₁₆	—	2	1	no	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88059	—	2	1	no	—
11	C₂×#88059	C₂²	1	2	yes	—
12	C₂²×#6090	C₂²	1	2	yes	—
13	#11718585	—	2	1	no	—
14	C₁₃×#341	—	2	1	no	—
15	#11718085	—	2	4	no	—
16	C₄⋊#6090	C₄	1	2	yes	—
17	C₄⋊#6090	C₄	1	2	yes	—
18	C₂×C₁₃×M₁₆	—	2	1	no	—
19	C₂×C₁₃×M₁₆	—	2	1	no	—
20	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
23	C₂×#6099	—	2	2	no	—
24	C₁₃×#108	—	2	1	no	—
25	C₁₃×#108	—	2	1	no	—
26	C₁₃×#108	D₄	1	2	yes	—
27	C₁₃×#108	Q₈	1	2	yes	—
28	C₁₃×#97	—	4	1	no	—
29	C₄×C₈×C₁₃	—	4	1	no	—
30	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
34	#88059	—	2	2	no	—
35	#88059	—	2	2	no	—
36	#88059	—	4	2	no	—
37	C₂×#6090	—	2	2	no	—
38	C₂×#6090	—	2	2	no	—
39	C₂×#6090	C₂×C₄	1	4	yes	—
40	C₄⋊#1055	—	4	1	no	—
41	C₄⋊#1055	—	4	1	no	—
42	#13179	—	13	1	no	maximal, Sylow
43	C₂²×C₄×C₁₃	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	Q₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₁₃×M₁₆	—	2	2	no	—
47	C₁₃×M₁₆	—	2	2	no	—
48	C₁₃×M₁₆	—	4	1	no	—
49	C₁₃×M₁₆	—	4	1	no	—
50	C₂×C₈×C₁₃	—	2	1	no	—
51	C₂×C₈×C₁₃	—	2	1	no	—
52	C₂×C₈×C₁₃	—	4	1	no	—
53	C₂×C₈×C₁₃	—	4	1	no	—
54	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
61	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
62	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
63	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
64	C₄²×C₁₃	D₈	1	1	yes	characteristic
65	C₄²×C₁₃	D₈	1	1	yes	characteristic
66	#6099	—	4	2	no	—
67	#6099	—	4	2	no	—
68	C₂×#1055	—	4	2	no	—
69	#6090	—	2	4	no	—
70	C₂×#368	—	13	2	no	—
71	C₂×C₄⋊C₁₆	—	13	1	no	—
72	#1848	—	13	1	no	—
73	C₄×#108	D₁₃	1	1	yes	characteristic
74	#1381	—	13	2	no	—
75	C₂³×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	4	1	no	—
78	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
88	C₈×C₁₃	—	2	1	no	—
89	C₈×C₁₃	—	2	1	no	—
90	C₈×C₁₃	—	4	2	no	—
91	C₈×C₁₃	—	4	2	no	—
92	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
93	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
94	C₈×C₁₃	#125	1	1	yes	characteristic
95	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
96	#1055	—	8	2	no	—
97	C₂²×C₁₆	—	13	2	no	—
98	C₂×#108	—	13	2	no	—
99	C₂×#108	—	26	1	no	—
100	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
101	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
102	#423	—	26	1	no	—
103	C₄×M₁₆	—	2	1	no	—
104	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
105	#368	—	26	4	no	—
106	C₄⋊C₁₆	—	13	2	no	—
107	C₄⋊C₁₆	—	13	2	no	—
108	#341	—	2	1	no	—
109	C₂²×C₁₃	—	2	1	no	—
110	C₂²×C₁₃	—	2	1	no	—
111	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
112	C₂²×C₁₃	#462	1	1	yes	characteristic
113	C₂²×C₁₃	#477	1	1	yes	characteristic
114	C₄×C₁₃	—	2	1	no	—
115	C₄×C₁₃	—	2	1	no	—
116	C₄×C₁₃	—	2	1	no	—
117	C₄×C₁₃	#368	1	2	yes	complemented
118	C₄×C₁₃	#389	1	1	yes	characteristic
119	C₄×C₁₃	#428	1	1	yes	characteristic
120	C₄×C₁₃	#459	1	1	yes	characteristic
121	C₄×C₁₃	#474	1	1	yes	characteristic
122	C₂×M₁₆	—	2	1	no	—
123	C₂×M₁₆	—	2	1	no	—
124	C₂×M₁₆	—	26	2	no	—
125	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
127	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
128	C₂×C₁₆	—	13	4	no	—
129	C₂×C₁₆	—	26	2	no	—
130	C₂×C₁₆	—	26	2	no	—
131	#108	—	2	1	no	—
132	#108	—	2	1	no	—
133	#108	—	26	2	no	—
134	#108	—	26	2	no	—
135	#108	—	52	2	no	—
136	#108	Dic₂₆	1	2	yes	—
137	#108	D₅₂	1	2	yes	—
138	C₄⋊C₈	—	52	1	no	—
139	C₄⋊C₈	—	52	1	no	—
140	#97	—	4	1	no	—
141	C₄×C₈	—	4	1	no	—
142	C₄×C₈	Dic₂₆	1	1	yes	characteristic
143	C₄×C₈	Dic₂₆	1	1	yes	characteristic
144	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
145	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
146	C₂×C₁₃	—	2	1	no	—
147	C₂×C₁₃	—	2	1	no	—
148	C₂×C₁₃	#1947	1	1	yes	characteristic
149	C₂×C₁₃	C₂×#368	1	1	yes	characteristic
150	C₂×C₁₃	#2161	1	1	yes	characteristic
151	C₂²×C₄	#6107	1	1	yes	characteristic
152	C₂²×C₄	#6107	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
154	M₁₆	—	2	2	no	—
155	M₁₆	—	2	2	no	—
156	M₁₆	—	4	1	no	—
157	M₁₆	—	4	1	no	—
158	M₁₆	—	52	2	no	—
159	M₁₆	—	52	2	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	52	2	no	—
165	C₂×C₈	#6101	1	1	yes	characteristic
166	C₂×C₈	#6101	1	1	yes	characteristic
167	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
168	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
169	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
170	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
171	C₂×C₈	#6127	1	1	yes	characteristic
172	C₂×C₈	#6127	1	1	yes	characteristic
173	C₄²	#6101	1	1	yes	characteristic
174	C₄²	#6101	1	1	yes	characteristic
175	C₄²	#6104	1	1	yes	characteristic
176	C₄²	#6104	1	1	yes	characteristic
177	C₁₆	—	26	4	no	—
178	C₁₃	#13179	1	1	yes	characteristic, complemented, Sylow
179	C₂³	C₂×#6107	1	1	yes	characteristic
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	#88047	1	1	yes	characteristic
183	C₂×C₄	#88047	1	1	yes	characteristic
184	C₂×C₄	#88047	1	1	yes	characteristic
185	C₂×C₄	#88047	1	1	yes	characteristic
186	C₂×C₄	C₂×#6101	1	1	yes	characteristic
187	C₂×C₄	#88178	1	1	yes	characteristic
188	C₂×C₄	#88184	1	1	yes	characteristic
189	C₂×C₄	C₂×#6104	1	1	yes	characteristic
190	C₂×C₄	#88203	1	1	yes	characteristic
191	C₂×C₄	#88205	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	4	2	no	—
195	C₈	—	4	2	no	—
196	C₈	—	104	2	no	—
197	C₈	C₄×Dic₂₆	1	1	yes	characteristic
198	C₈	#88123	1	1	yes	characteristic
199	C₈	#88124	1	1	yes	characteristic
200	C₈	C₄×D₅₂	1	1	yes	characteristic
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₂×#88047	1	1	yes	characteristic
204	C₂²	#11718682	1	1	yes	characteristic
205	C₂²	#11718692	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	#11718085	1	2	yes	complemented
210	C₄	#11718589	1	1	yes	characteristic
211	C₄	#11718656	1	1	yes	characteristic
212	C₄	#11718668	1	1	yes	characteristic
213	C₄	#11718676	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	#13115594	1	1	yes	characteristic
217	C₂	C₂×#11718085	1	1	yes	characteristic
218	C₂	#13116630	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#6104
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×#108	C₂
socle	C₂²×C₁₃	C₂×#107
prime radical	C₂³×C₁₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×#108	D₁₃
hypocenter	C₁₃	#13179
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
117. C₄×C₁₃	#368	semidirect
178. C₁₃	#13179	semidirect
209. C₄	#11718085	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#11718085	2
C₂×C₄⋊#6090	2
#13115008	2
#13112929	2
C₄×C₁₃×#108	2
#13179	13